A QMC Study of the Homogeneous Electron Gas

Graham Spink and Richard Needs, TCM, University of Cambridge

Homogeneous Electron Gas

bbbei ij jii

iHHH

ˆˆ1

2

1

2

1ˆ 2

rr

•Electrons move in uniform, positive background potential:

•Used in Local Density Approximation in DFT:

•Only one independent parameter, rs, the “typical” distance between electrons. ζ is the spin polarization.

rrr dnnnXC

LDA

XCE ))(()(][

Pair Correlation Function

RR

RrrR

d

dr

Nrrg 2

21

2

2)(

)()(1

4)(

1

21

21

Momentum Density

R

xRrrrrxrr

k

xk

d

dde

rn

N

i Ni

Nii

s 21 1

12

),,,,(),,,,(

),(

Gori-Giorgi and Ziesche, PRB 66, 235116

“New” Techniques• More accurate Jastrow factor• Backflow

López Ríos et al., submitted to Phys. Rev. E;

@rs=0.5 increases VMC correlation energy by 5%

• Three-body Jastrow terms Holtzmann et al., Phys. Rev. E 68 046707; @ rs=20, increases VMC correlation energy by over 1%

• Different treatment of finite-size effects Drummond et al.,

PCF Size Convergence

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5

r/rs (a.u.)

Anti-

para

llel s

pin

PCF

rs0.5,118e

rs0.5, 226e

rs20, 118e

rs20, 226e

PCF with Backflow

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5r/rs

anti-

para

llel P

CF

rs0.5;noBF

rs0.5;BFrs20;noBF

rs20;BF

rs3;noBFrs3;BF

Momentum Densityrs=3.0, z=0

-2.00E-01

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00

1.20E+00

0.00E+00 2.00E-01 4.00E-01 6.00E-01 8.00E-01 1.00E+00 1.20E+00 1.40E+00 1.60E+00

q/kF

n(q

)

Ortiz & Ballone

CASINO

PCF at rs=0.5 in VMC and DMC

0.6

0.7

0.8

0.9

1

1.1

0 0.5 1 1.5 2 2.5

r/a.u.

Anti-

para

llel P

CF

DMC

VMC

Pair Correlation Function

RR

RrrR

d

dr

Nrrg 2

21

2

2)(

)()(1

4)(

1

21

21

On top pair density

DFT Finite Size Corrections

Conclusions and Future Work

• New techniques in QMC simulations make significant difference to quantities considered

• With new data we can test recently proposed fitting procedures

• Now continuing the above at intermediate densities and with DMC

Results for PCF

0.000000

0.100000

0.200000

0.300000

0.400000

0.500000

0.600000

0.700000

0.800000

0.900000

1.000000

1.100000

0.000000 1.000000 2.000000 3.000000 4.000000 5.000000 6.000000 7.000000 8.000000

r/a.u.

Without Backflow

With Backflow

PC

F (

anti-

para

llel s

pins

)

VMC, 162 electrons, rs=3.0, ζ=0.0